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Theorem csb2 3189
Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that  x can be free in  B but cannot occur in  A. (Contributed by NM, 2-Dec-2013.)
Assertion
Ref Expression
csb2  |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
Distinct variable groups:    x, y, A    y, B
Allowed substitution hint:    B( x)

Proof of Theorem csb2
StepHypRef Expression
1 df-csb 3188 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbc5 3121 . . 3  |-  ( [. A  /  x ]. y  e.  B  <->  E. x ( x  =  A  /\  y  e.  B ) )
32abbii 2492 . 2  |-  { y  |  [. A  /  x ]. y  e.  B }  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
41, 3eqtri 2400 1  |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2366   [.wsbc 3097   [_csb 3187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-sbc 3098  df-csb 3188
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